Toward a non-commutative Gelfand duality: Boolean locally separated toposes and Monoidal monotone complete $C^{*}$-categories

0.36 MB·English

by  Simon Henry

Toward a non-commutative Gelfand duality:
Boolean locally separated toposes and Monoidal
C
monotone complete -categories
∗
5
1 Simon Henry
0
2 January 29, 2015

n
a
J
Abstract

8
2 **DraftVersion**Toanybooleantoposonecanassociateitscategory
of internal Hilbert spaces, and if the topos is locally separated one can

] consider a full subcategory of square integrable Hilbert spaces. In both
T caseitisasymmetricmonoidalmonotonecompleteC∗-category. Wewill
C
provethatanybooleanlocallyseparatedtoposcanbereconstructedasthe
h. classifying topos of “non-degenerate” monoidal normal ∗-representations
t of both its category of internal Hilbert spaces and its category of square
a
integrable Hilbert spaces. This suggest a possible extension of the usual
m
Gelfanddualitybetweenaclassoftoposes(ormoregenerallylocalicstacks
[ orlocalicgroupoids)andaclassofsymmetricmonoidalC∗-categoriesyet

tobe discovered.
1
v
5
4 Contents
0
7
0
1 Introduction 2
.
1
0 2 General preliminaries 3
5
1
3 Monotone complete C∗-categories and boolean toposes 6
:
v
i 4 Locally separated toposesand square integrable Hilbertspaces 7
X
r
5 Statement of the main theorems 9
a
6 From representations of red to geometric morphisms 13
H
6.1 Construction of the geometric morphism on separating objects . 13
6.2 Functoriality on separating object . . . . . . . . . . . . . . . . . 20
6.3 Construction of the Geometric morphism . . . . . . . . . . . . . 22
6.4 Proof of the “reduced” theorem 5.8 . . . . . . . . . . . . . . . . 25
Keywords. Boolean locally separated toposes, monotone complete C*-categories, recon-
structiontheorem.
2010 Mathematics Subject Classification. 18B25,03G30, 46L05,46L10.
email: [email protected]
1
7 On the category ( ) and its representations 26
H T
7.1 The category ( ) . . . . . . . . . . . . . . . . . . . . . . . . 26
/X
H T
7.2 Tensorisationby square integrable Hilbert space . . . . . . . . . 29
7.3 Proof of the “unreduced” theorem 5.5. . . . . . . . . . . . . . . . 30
8 Toward a generalized Gelfand duality ? 31
1 Introduction
This is a draft version. It will be replaced by a more definitive version within a
few months. In the meantime any comments is welcome.
Thispaperispartofaprogram(startingwiththeauthor’sphdthesis)devotedto
the study of the relation between topos theory and non-commutative geometry
astwogeneralizationsoftopology. The centraltheme ofthis researchprojectis
the construction explained in section 2 which naturally associate to any topos
the C∗-category ( ) of Hilbert bundles over .
T H T T
Inthepreviouspaper[7]westudied“measuretheory”oftoposesandcompareit
tothetheoryofW∗-algebras(vonNeumannalgebras)throughthisconstruction
of ( ). At the end of the introduction of this previous paper one can find a
H T
table informally summing up some sort of partially dictionary between topos
theoryandoperatoralgebratheory. Thegoalofthepresentworkissomehowto
providea frameworkformakingthis dictionaryaconcretemathematicalresult,
byshowingthatabooleanlocallyseparatedtopos canactuallybecompletely
reconstructed from any of the two symmetric moTnoidal1 C∗-categories ( )
H T
and red( ) of Hilbert bundles and square-integrableHilbert bundles.
H T
More precisely, we will show that if is a boolean locally separated topos,
T
then is the classifyingtopos fornon-degenerate2 normalsymmetricmonoidal
T
representations of either red( ) and ( ). These are not geometric theory,
H T H T
notevenfirstordertheoryinfact,butwewillproveanequivalenceofcategories
betweenpoints of andthese representationsoveranarbitrarybase topos,see
T
theorems 5.5 and 5.8 for the precise statement.
Sections2,3and4containsomepreliminarieswhicharemostly,butnotentirely,
recall of previous work.
Section5containsthestatementofthetwomaintheoremsofthepresentpaper:
theorem5.5,whichisthereconstructiontheoremfromthe“unreduced”category
( ) andtheorem 5.8 whichis the reconstructiontheoremfromthe “reduced”
H T
category red( ) of square integrable Hilbert spaces.
H T
Sections 6 and 7 contain respectively the proof of the “reduced” theorem 5.8
and the proof that the reduced theorem imply the unreduced theorem 5.5.
Thekeyarguments,whichactuallyusethespecificityofthehypothesis“boolean
locally separated” seems to all be in subsection 6.1, and the rest of the proof
1Thetermmonoidalistakenwithaslightlyextendedmeaning: Hred(T)hasingeneralno
unitobject
2seedefinitions 5.4and5.7.
2
seems to us, at least in comparison, more elementary (it is mostly about some
computationsofmultilinearalgebra)andalsomoregeneral(thespecifichypoth-
esis are essentially3 not used anymore). In fact, we have unsuccessfully tried to
prove a result in this spirit for several years, and results of subsection 6.1 have
always been the main stumbling block.
Finally we conclude this paper with section 8, where we explain how the re-
sults of this paper might maybe be extended into a duality between certain
monoidal symmetric C∗-categories and certain geometric objects (presumably,
localic groupoids or localic stacks) which would be a common extension of
the usual Gelfand duality, the Gelfand duality for W∗-algebras, the Doplicher-
Robertsreconstructiontheoremforcompactgroups,andofcoursetheresultsof
the present paper. This would somehow constitute a sort of non-commutative
Gelfand duality. Of course the existence of such a duality, and even its precise
statementareatthepresenttimehighlyconjectural,butwewilltrytohighlight
what are the main difficulties on the road toward such a result.
This conjectured duality is formally extremely similar to the reconstruction
theorems obtained for algebraic stacks, as for example [11], [3].
2 General preliminaries
We will make an intensive use of the internal logic of toposes (i.e. the Kripke-
Joyal semantics for intuitionist logic in toposes) in this paper. A reader unfa-
miliar with this technique can read for example sections 14,15 and 16 of [13]
which give a relatively short and clear account of the subject. Other possible
references are [2, chapter 6], [12, chapter VI], or [10, D1 and D4].
Because this paper is mostly about boolean toposes and monotone complete
C∗-algebras we will assume that the base topos (that is the category of set)
satisfies the law of excluded middle, but we won’t need to assume the axiom of
choice.
It is reasonable to think that these results can also be formulated and proved
over a non-boolean basis, but the gain in doing so would be very small: over a
non-boolean basis, any boolean topos and any monotone complete C∗-algebra
is automatically defined over a boolean sub-locale of the terminal object, and
hence can be dealt with in a boolean framework.
This being said, a large part of the proofs will take place internally in a non
boolean topos and hence will have to avoidthe law of excluded middle anyway.
We will also often have to juggle between internal and external logic or be-
tweenthe internallogic oftwo differenttoposes. We will generallyprecisewhat
statement has to be interpreted “internally in ” or “externally”, but we also
T
would like to emphasis the fact that most of the time the context makes this
completely clear: if an argument start by “let x X” and that X is not a set
∈
but anobjectofa topos itobviouslymeansthat weare workinginternallyin
T
3Lemma6.3.2seemstobetheuniqueexception
3
. This convention,is in fact completely similar with the usual use of context4
T
in mathematics: when a mathematicians sayssomething like “letx S” (for S
∈
aset)thenwhatfollowsisactuallymathematicsinternalto thetoposSets of
/S
sets overS, indeed everything being said after implicitly depends on a parame-
ter s S, and if the conclusion actually does not depends on s then it will be
∈
valid independently of the context only if S was non-empty.
Our convention is hence that when we say something like “let x X” where
∈
X is an object of a topos then what follows is internal to the topos , or
/X
T T
equivalently, internal to with a declared variable x, and this being true until
T
the moment where this variable x is “removed” from the context (in classical
mathematics, this is usually left implicit because, as soonas X is non-empty, it
is irrelevant,but in ourcase we willgenerallymake itprecise by sayingthat we
are now working externally).
Wewillofcoursesayexplicitlyinwhichtoposweareworkingassoonaswethink
that it actually improve the readability, but we also think that this perspective
makes(onceweareusedtoit)thechangefromworkinginternallyintoonetopos
from another topos as simple as introducing and forgetting abstract variables
in usual mathematics and makes the text easier to read.
Allthe toposesconsideredareGrothendiecktoposes,inparticularthey allhave
a natural numbers object (see [10, A2.5 and D5.1]) “N” or “NT” which is just
the locally constant sheaf equal to N.
Aset,oranobjectX ofatopos,issaidtobeinhabited if(internally)itsatisfies
x X. For an object of a topos it corresponds to the fact that the map from
∃ ∈
X to the terminal object is an epimorphism.
An object X of a topos is said to be a bound of if subobjects of X form
T T
a generating family of (i.e. is any object of admit a covering by subobject
T T
of X). Every Grothendieck topos admit a bound (for exemaple take the direct
sumofalltherepresentablesheavesforagivensiteofdefinition), infacttheex-
istenceofaboundtogetherwiththe existenceofsmallco-productscharacterize
Grothendieck toposes among elementary toposes.
Let beanarbitrarytopos. CT istheobjectof“continuous5complexnumbers”
T
thatisRT RT whereRT istheobjectofcontinuous/Dedekindrealnumbersas
×
defined forexample in [10, D4.7]. Inany topos (with a naturalnumber object),
CT is a locale ring object.
WhenP isadecidableproposition,i.e. ifonedoeshaveP ornot-P,onedenotes
I the realnumber defined by I =1 if P and I =0 otherwise. An objectX is
P P P
saidtobe decidable ifforallxandy inX the propositionx=y isdecidable,in
which case we denote δ for I . i.e. δ is one if x=y and zero otherwise.
x,y x=y x,y
By a “Hilbert space of ”, or a -Hilbert space we mean an object H of ,
T T T
endowedwithaCT-modulestructureandascalarproductH H CT (linear
× →
4“Context”istakenhereinitsformal“typetheoretic”meaning,i.e. the“set”ofallvariable
thathasbeendeclaredatagivenpointofaproof.
5alsocalledDedekindcomplexnumbers
4
in the second variable and anti-linear in the first), which satisfies internally all
the usual axioms for being a Hilbert space, completeness being interpreted in
term of Cauchy filters, or equivalently Cauchy approximations but not Cauchy
sequences.
( )denotestheC∗-category6whoseobjectsareHilbertspacesof andwhose
HmoTrphisms are “globally bounded operators”, that is linear mapsTf : H H′
→
whichadmitanadjointandsuchthat itexistsanexternalnumber K satisfying
(internally)forallx H, f(x) 6K x . Thenorm f ∞ isthenthesmallest
∈ k k k k k k
such constant K (if we were not assuming the law of excluded middle in the
basetopositwouldbeanuppersemi-continuousrealnumber),theadditionand
composition of operators is defined internally , f∗ is the adjoint of f internally,
and this form a C∗-category.
Because the tensor product of two Hilbert spaces can easily be defined, even
in intuitionist mathematics, the category ( ) is endowed with a symmetric
H T
monoidal structure.
Precise definition of a symmetric monoidal category, a symmetric monoidal
functor, and a symmetric monoidal natural transformation can be found in
S.MacLane’s category theory books7 [15], chapter XI, section 1 and 2.
Briefly,asymmetric monoidalcategoryis a categoryendowedwitha bi-functor
, a specific “unit” object e and isomorphisms e A A e A,
⊗ ⊗ ≃ ⊗ ≃
(A B) (B A)and(A B) C A (B C)whicharenaturalandsatisfies
⊗ ≃ ⊗ ⊗ ⊗ ≃ ⊗ ⊗
certain coherence conditions. A symmetric monoidal functor (“braided strong
monoidal functor” in MacLane’s terminology) is a functor between symmetric
monoidal categories with a natural isomorphism F(A B) F(A) F(B)
⊗ ≃ ⊗
which has to satisfy a certain number of coherence and compatibility relations.
Finally a symmetric monoidaltransformation(the adjective symmetric is actu-
ally irrelevant for natural transformation) is a natural transformation between
symmetric monoidal functor which satisfy coherence conditions, stating that,
uptothe previouslydefinednaturalisomorphisms,ηA ηb isthe sameasηA⊗B
⊗
and η is the identity.
e
Moreover, when we are talking about (symmetric) monoidal C∗-categories or
symmetric monoidal -functor between such categories we are alwaysassuming
∗
that all the structural isomorphisms are in fact isometric isomorphisms, i.e.
their inverse is their adjoint. For example, it is clearly the case for ( ).
H T
Finally, if f : is a geometric morphism between two toposes, and H is a
Hilbert spaceEof→Tthen f∗(H) is a “pre-Hilbert” space of but fails in general
T E
tobecompleteandseparated,wedenotebyf♯(H)its separatedcompletion. f♯
is a symmetric monoidal -functor from ( ) to ( ).
∗ H T H E
6seeforexemple[6]forthedefinitionofC∗-category.
7Onlyinthesecondedition(1998).
5
3 Monotone complete C -categories and boolean
∗
toposes
We recall that a C∗-algebra is said to be monotone complete if every bounded
directed net of positive operators has a supremum. A positive linear map be-
tween two monotone complete C∗-algebras is said to be normal if it preserves
supremum of bounded directed set of positive operators.
The theory of monotone complete C∗-algebras is extremely close to the theory
of W∗-algebras, in fact it is well know that a monotone complete C∗-algebra
having enough normal positive linear form is a W∗-algebra (see [16, Theorem
3.16]).
When is a boolean topos, ( ) is a monotone complete C∗-category in
the senTse that it has bi-productHs aTnd the C∗-algebra of endomorphisms of any
object is monotone complete. Indeed, because is boolean, the supremum of
T
a bounded net of operators can be computed internally, and as the supremum
is unique it “patches up” into an externally defined map, see [7, section 2].
MonotonecompleteC∗-categoriesareextremelyclosetotheW∗-categoriesstud-
iedin[6],infactmostofthe resultof[6]whichdoesnotinvolvethe existenceof
normalstates (or the modular time evolution)alsoholdfor monotonecomplete
C∗-categories. We will review some of these results:
If C is a monotone complete C∗-category then we define the center Z(C) of C
as being the commutative monotone complete C∗-algebraof endomorphisms of
the identity functor of C. In the more general situation Z(C) might fail to be
a setand be a properclass,but we will notbe concernby this issue because we
proved in [7, 3.6] that ( ) has a generator and hence, by results of [6], the
H T
algebraZ(C)canbe identifiedwiththe centerofthe algebraofendomorphisms
of this generator.
IfA is anobjectofa monotonecomplete C∗-categorythenwe define its central
support c(A) Z(C) by:
∈
c(A) := sup ff∗ Hom(B,B).
B
f:An→B ∈
kfk<1
If the monotone complete category we are working with admit bi-product then
An denotes the bi-product of n-copies of A and if not, then one can still make
sense of a map (f ,...,f ) from An to B as the data of n maps from A to B,
1 n
ff∗ is f f∗ a,d f isdefinedas ff∗ 1/2. Onecancheckthatthesupremum
P i i k k k k
involved in the definition of C(A) is directed by showing that the set of such
B
“ff∗”isinorderpreservingbijectionwiththesetofff∗ wheref isanarbitrary
mapsfromAn toB withoutconditiononthenorm,thebijectionbeingobtained
by multiplying f by a convenient function of ff∗.
Equivalently,c(A)canbedefinedasthesmallestprojectioncinZ(C)suchthat
c =Id , but we will need the fact that it is a directed supremum.
A A
OnesaysthatanobjectAisquasi-contained inanobjectB ifc(A)6c(B). An
object A is said to be a generator of a monotone complete C∗-category if and
only if c(A)=1 i.e. if every other object is quasi-contained in A.
6
One can for example check that if two normal functors agree on a generator
and its endomorphisms then they are isomorphic: it is an easy consequence
of results of [6] for W∗-categories and the proof can extended to monotone
complete C∗-categories easily.
We conclude this section by briefly mentioning what quasi-containement mean
in the case of ( ):
H T
3.1. Proposition : Let be a boolean topos and H,H′ ( ) two Hilbert
spaces of , then H is weTakly contained in H′ if and only∈ifHtheTre exists a set
F of bounTded operators from H′ to H such that internally in the functions in
T
F spam a dense subspace of H.
If this is true, we will say that H is covered by the maps in F.
Proof :
IfH isweaklycontainedinH′ thenc(H)6c(H′)hencec(H′) =1. Rewriting
H
this using the definition of c(H′) one gets:
Id = sup ff∗ Hom(H,H)
H
f:(H′)n→H ∈
kfk<1
Butaswementionedearlier,supremumsofdirectednetsin ( )arecomputed
H T
internally,thismeansthatthissupremumconvergeinternallyforthestrongop-
eratortopology. Inparticular,foranyh H onehasff∗(h)whichisarbitrarily
∈
close to h when f run through the (external) set:
F = f f :(H′)n H, f <1
0
{ | → k k }
Hence taking F to be the set of “component” of maps in F , the sum of the
0
images of maps in F spam all of H.
Conversely, assume that H is spammed by a family F of external maps f :
H′ H. For any f : H′ H, and in particular, for any f F one has
c(H→′) f =f c(H′) =f→. The projector c(H′) is hence (int∈ernally in )
H H′ H
◦ ◦ T
equalto the identity on the imageof allthe maps f F and hence on allofH,
i.e. c(H′) =Id which proves that c(H)6c(H′).∈(cid:3)
H H
4 Locally separated toposes and square integrable
Hilbert spaces
We recall (see [14, chapter II]) that a topos is said to be separated if its
T
diagonal map (which is localic by [10, B3.3.8]) is proper, i.e. if,
T → T ×T
when seen as a ( )-locale though the diagonal maps, is compact.
T ×T T
In [7, theorem 5.2] we provedthat a boolean topos is separatedif and only if it
is generated by internally finite objects. One will use a slightly modified form
of this result :
7
4.1. Theorem : Let bea boolean separated topos, then admit a generating
T T
family of objects (X) such that for each X there exists an interger n such that
internally in , the cardinal of X is smaller than n.
T
One will say that such objects are of bounded cardinal.
Proof :
Thistheoremisanimmediateconsequenceof[7,theorem5.2]: isgeneratedby
T
a familly of internally finite object X, but for each object X of this generating
familly and for each natural number n one can define X := x X X 6 n
n
{ ∈ || | }
whose cardinal is internally bounded by n, and as X is internally finite the
(Xn)n∈N form a coveringfamily of X andhence the (Xn)X,n forma generating
family fulfilling the property announced in the theorem. (cid:3)
An object X of boolean topos is said to be separating if the slice topos
/X
T
is separated. A boolean topos is said to be locally separated if it admit an
inhabited separating object, or equivalently if any object can be covered by
separating objects, see [7] section 5 for more details.
If X is an object of a boolean topos then one can define the -Hilbert space
T T
l2(X) of square sumable sequences indexed by X. It can also be done in a non
boolean topos but it require X to be a decidable8 object. Internally in , the
T
space l2(X) has generators e for x X such that e ,e =δ .
x x y x,y
∈ h i
4.2. Proposition: LetX beaboundofabooleantopos andY beaseparating
T
object of then l2(Y) is quasi-contained in l2(X).
T
Proof :
Let X be a bound and Y any separating object. As X is a bound, Y can be
covered by maps h:U Y with U X. Using the fact that is separated
/Y
→ ⊂ T
and boolean, we know that it is generated by objects of bounded cardinal and
the imageofa mapwhosedomainis finite isalsofinite, andofsmallercardinal,
hence U admit a covering by sub-objects with bounded cardinal in , hence
/Y
T
we can freely assume that U is itself of bounded cardinal in .
/Y
T
One can then define (for each such map h:U Y) a map φ :l2(X) l2(Y)
h
→ →
by φ (e ) = 0 if x / U and φ (e ) = e if x U. The fact that the object
h x h x h(x)
∈ ∈
U is finite with bounded cardinal over Y mean that there exists an (external)
integer n such that each fiber of h has cardinal smaller than n, this is exactly
what we need to know to construct the adjoint of φ and to prove that φ is
h h
bounded (and hence extend into an operator).
Nowassuchmapsh:U Y coverY,onehasinternallyin : “ y Y, h H
→ T ∀ ∈ ∃ ∈
such that y is in the image of h”, where H denote the external set of such map
h : U Y which are finite and of bounded cardinal in . In particular the
/Y
→ T
jointimageofalltheφ forh H containsallthegeneratorsofl2(Y)andhence
h
∈
8Inordertodefinethescalarproductoftwogeneratorsortodefinethesumofasequence
weneedthatforanyx,y∈X x=y orx6=y.
8
spam a dense subspace of l2(Y), which concludes the proof by proposition 3.1.
(cid:3)
Wedenoteby red( )thefullsubcategoryof ( )ofobjectswhichareweakly
H T H T
contained in l2(X) for some separating object X. Objects of red( ) are said
H T
to be square-integrable9.
Results of [7] (especially section 7) suggest that the square integrable Hilbert
spaces of are the one that are clearly related to the geometry of .
T T
Because of proposition 4.2, for any separating bound X of the Hilbert space
T
l2(X)isageneratorof red( ),henceextendingproposition7.6of[6]tomono-
tonecompleteC∗-categHory(oTrrestrictingourselvestotoposwhichareintegrable
in the sense of[7, section 3])gives us that when is a booleanand locally sep-
T
aratedtopos, red( )isequivalentto thecategoryofreflexiveHilbertmodules
H T
over End(l2(X)). For this reason we can call this algebra “the” (or “a”) re-
duced10 algebra of (it is unique up to Morita equivalence).
T
Finally, The fact that it is possible to define internally the tensor product of
two Hilbert spaces yields a symmetric monoidal structure on ( ). Moreover
H T
asl2(X) l2(Y) l2(X Y)onecanseethat red( )isstablebytensorprod-
⊗ ≃ × H T
uct11. Hence, red( ) is also endowed with a symmetric monoidal structure,
H T
but without a unit object (unless is separated).
T
5 Statement of the main theorems
5.1. Definition: Let C beamonotone completeC∗-category, anytopos (not
E
necessary boolean) a representation of C in is a -functor ρ from C to ( ).
E ∗ H E
It is said to be normal if for any supremum a=supa of a bounded directed net
i
of positive operator in C, ρ(a )convergeinternally in theweakoperator topology
i
to ρ(a).
A representation of C in is also the same as a representation of p∗C in the
E
category of Hilbert space internally in (where p is the geometric morphism
E
from to the point). Also if is boolean, then ( ) is monotone complete
E E H E
and a representation ρ is normal in the sense of this definition if and only if it
is normal as a C -functor between C∗-category.
∗
9because when T is the topos of G-sets for some discrete group G, then H(T) is the
category of unitary representations of G while Hred(T) is precisely the category of square
integrablerepresentations ofG
10For example, if T is the topos of G-sets for G a discrete group one obtains the usual
(reduced) vonNeumannalgebraofthegroupthisway.
11in7.2.2wewillactuallyprovethestrongerresultthatthetensorproductofanarbitrary
HilbertspacewithasquareintegrableHilbertspaceissquareintegrable
9
5.2. When C has additional structure (for example is monoidal) we will by
defaultassumethatthe representationρpreservethese structures,forexemple:
Definition : If is a boolean topos and an arbitrary topos, a representation
of ( ) in is aTnormal representation oEf the monotone complete C∗-category
H T E
( ) in such that the underlying -functor is symmetric and monoidal.
H T E ∗
5.3. Definition: Wewill say thata Hilbert spaceH ( ) is inhabited if one
∈H E
has s H suchthat s >0internallyin , or equivalently, if s H, s =1
∃ ∈ k k E ∃ ∈ k k
holds internally in .
E
5.4. Definition : If is a boolean topos and an arbitrary topos, a represen-
T E
tation ρ of ( ) is said to be non-degenerate if for any inhabited object H of
H T
( ) the Hilbert space ρ(H) is inhabited in .
H T E
It is important to notice that detecting whether a representation ρ of ( ) is
H T
non-degenerate or not can be done completely from the (monoidal) category
( ) without knowing the topos . Indeed, an object H of ( ) is inhabited
H T T H T
in if and only if the functor H is faithful.
T ⊗
Asanexampleofa“degenerate”representation,onecanconsider thetoposof
T
G-setsforsomeinfinite discretegroupeG, then ( )isthecategoryofunitary
H T
representation of G. Let ρ be the representation of ( ) into defined by:
H T T
ρ(H)= h H h belongs to a finite dimensional sub-representation
{ ∈ | }
One has ρ(l2(G)) = 0 so it cannot be non-degenrate. One easily checks that
it is a symmetric normal -functor, and it is monoidal because of the following
∗
observation:
Lemma : Let R,R′ ( ) be two representations of G. Assume that R R′
∈H T ⊗
contains a (non trivial) finite dimensional sub-representation, then both R and
R′ contains a non trivial finite dimensional sub-representation.
Proof :
Let K R R′ a non-trivial finite dimensional sub-representation. let K∗ the
⊂ ⊗
dual of the representation K. Because K is finite dimensional and non-trivial
K K∗ containsanon-zeroinvariantvector. InparticularR R′ K∗ contains
⊗ ⊗ ⊗
a non-zero invariant vector which corresponds to a non zero G-linear Hilbert-
Schmidt f from R to R′∗ K, f∗f is hence a non-zero compact self-adjoint
⊗
G-linearautomorphismofRwhichishencegoingtohavesomenon-trivialfinite
dimensional G-stable eigenspaces. This concludes the proof of the lemma. (cid:3)
10